Local Stability of Disk Galaxies

Astronomy 626: Spring 1995

The existence of an equilibrium solution to the CBE does not assure
its stability. Real stellar systems are subject to perturbations, and
if these grow they may completely transform the initial equilibrium.
Thus it is critical to check that our galaxy models are actually
stable.

Physics of the Jeans Instability

The Jeans (1929) instability is probably the most basic instability
in gravitating systems. It was originally derived to discuss a
since-abandoned model for the formation of the Solar System, but it is
now recognized as of fundamental importance for cosmology.

To understand the Jeans instability, consider a nearly-uniform
distribution of stars containing a slightly overdense spherical
region with radius L and density rho. The
overdense region will collapse if the random velocities of stars are
not large enough to carry them out of the region before the collapse
can occur (e.g. Jeans 1929, Toomre 1964).

The collapse timescale can be estimated by considering a star at
rest on the edge of the sphere. The gravitational acceleration of
this star is just GM/L^2, where the mass of the sphere is

4pi 3
(1) M = --- L rho .
3

Then the time for this star to reach the center is just half the
Keplerian period for an orbit with semimajor axis L/2 about a
point-mass M, or

The timescale for stars to escape the overdense region is of order
L divided by the r.m.s. stellar velocity, or

(3) t_esc = L / v_rms .

Notice that t_coll is independent of L, while
t_esc increases linearly with L. Thus small regions
have t_esc &lt t_col and are stable, while large regions have
t_esc &gt t_col and are unstable. The critical radius where
collapse is just possible can be estimated by setting t_esc =
t_col; the result is the Jeans length,

3 pi v_rms^2 1/2
(4) L_J = (------------) .
32 G rho

Only those overdense regions with L &gt L_J are subject to
the Jeans instability.

Physics of Disk Instabilities

To form a physical understanding of disk instabilities, we can first
consider the Jeans instability in a non-rotating disk, and then consider
separately the effects of rotation (Toomre 1964, GKvdK89, Ch. 9.5).

Nonrotating disks

A 2-D version of the Jeans instability serves as a model for local
gravitational instability in a stellar disk. Let Sigma be
the surface density of the disk, and suppose that there is a region,
of radius L and mass

2
(5) M = pi L Sigma ,

which is slightly overdense. Approximating the collapse time by the
same Keplerian formula as above, we have

while the escape time is again given by Eq. (3). Notice that
t_coll is now proportional to L^1/2; because this is
still less than the linear proportionality of t_esc, the
reasoning used in the 3-D case applies here too. Thus by setting
t_esc = t_coll we obtain the Jeans length in 2-D

pi v_rms^2
(7) L_J = -- ------- .
8 G Sigma

As before, only overdense regions with L &gt L_J can collapse
before they are erased by random motions of stars.

Rotating disks

In a differentially rotating disk the local angular velocity is
Oort's constant B. A circular region collapsing from radius
L to radius L_1 will conserve its angular momentum, so
its angular velocity is

2 2
(8) Omega = B L / L_1 .

If we analyze the motion of the region in a frame of reference rotating
with angular velocity Omega we must include an outward-directed
pseudo-acceleration (centrifugal force), which at the edge of the disk
is

2 2 4 3
(9) a_r = L_1 Omega = B L / L_1 .

There is also an inward acceleration due to gravity:

2
(10) a_g = - ~ G M / L_1 ,

where once again a point-mass approximation has been used. Now the key
idea is that collapse will not occur if a_r initially
increases faster than a_g (Toomre 1964). This establishes
a maximum radius for collapse, because rotation is more important
on larger scales. Setting

3. Plug these perturbed solutions into the CBE and Poisson
Equation, and keep only terms of O(epsilon). This yields
linearized forms of these equations (see BT87, Ch. 5).

4. Solve the linearized equations to find the time-development
of an initial f_1(x,v,0). If any initial perturbation
can be shown to grow with time, the system is unstable. To prove
stability one must, in principle, consider all possible perturbations,
and show that none lead to growing solutions.

Local analysis: If the equilibrium solution is spatially
homogeneous, or if the characteristic length-scale of the
perturbations is much smaller than the characteristic length-scale of
the system (WKB approximation), the imposed perturbations can be
Fourier-analyzed in space and time into components of the form

i(k.x - omega t)
(16) f_1(x,v,t) = f_a(v) e ,

where k is the wave-number and omega is the frequency
of the perturbation. If any growing solutions of the linearized
CBE exist then there must be solutions of the form in Eq. (16) which
also grow, since any solution can be expressed as a sum of these Fourier
components. When Eq. (16) is inserted into the linearized CBE and
Poisson Equations, the result is a dispersion relation between
omega^2 and k. If omega^2 &lt 0 for any
value of k then perturbations with that wave-number are
unstable because then omega = i gamma for some real
gamma, and the corresponding Fourier component grows like

-i omega t gamma t
(17) f_1 ~ e = e .

Results of WKB analysis for disks

The WKB analysis of a differentially-rotating disk galaxy is covered
in BT87, Ch. 6.2. Here I will only quote results for axisymmetric
perturbations, which locally have the form

i(k R - omega t)
(18) f_1 ~ e ;

it turns out that such perturbations are sufficiently general to expose
the most important physical effects. The dispersion relations resulting
from such perturbations involve a quantity not yet mentioned: the radial
or epicyclic period of a star on a nearly circular orbit,

d Omega^2 1/2
(19) kappa = (R ---------+ 4 Omega^2) ,
dR

where Omega(R) is the angular velocity of the circular orbit at
radius R.

For a gas disk, the dispersion relation is

2 2 2
(20) omega = kappa - 2 pi G Sigma |k| + k v_s ,

where v_s is the speed of sound in the gas.

For a stellar disk, the dispersion relation depends on the
detailed form of the distribution function. If the random stellar
velocities in the disk are assumed to have a gaussian distribution, the
dispersion relation is

where sigma_R is the radial velocity dispersion and the
reduction factorF(s,chi) is given in Eq. (6-45) of
BT87. Note that F(s,0) = 1; thus Eqs. (20) and (21) are
identical in the limiting case where v_s = sigma_R = 0. This
is reasonable since the dynamical stability of a perfectly `cold' disk
should not depend on its make-up.

In either case, local stability against axisymmetric perturbations is
assured if omega^2 &gt 0 for all values of k. This
condition implies that

Q Parameters of Real Galaxies

An estimate of Q for the solar neighborhood is given in
BT87, Ch. 6.2. For the solar neighborhood, the surface density and
epicyclic period are roughly

2
(24) Sigma = 75 M_solar / pc ,
(25) kappa = 36 km / s / kpc ,

and the radial velocity dispersion, averaged over the vertical extent of
the disk, is

(26) sigma_r = 45 km / s .

To account for the finite thickness and gas content of the galactic
disk, the coefficient of 3.36 in Eq. (23) should be reduced to
~2.9 (Toomre 1974). The result is that for the solar
neighborhood

(27) Q = ~1.7 ,

so it appears that the Milky Way is locally stable.

For other disk galaxies, the radial dispersion profile may be
estimated by comparing gaseous and stellar rotation velocities; the
latter lag the former by an amount proportional to sigma_R^2
due to asymmetric drift. The few galaxies which have been studied so
far yield Q = 1.5 to 2; moreover, Q appears
to be fairly independent of the radius R (GKvdK89, Ch.
10.2).

It is easy to understand why Q &gt 1; if galactic disks
were locally unstable to gravitational collapse then massive clumps of
stars would form and scatter other stars, increasing the velocity
dispersion until Q = 1 was reached. But the actual
mechanism(s) responsible for randomizing the velocities of disk stars
are not completely understood. Scattering by giant molecular clouds
(which may represent gravitationally-collapsed clumps in the
gaseous disk) can explain part of the velocity increase, but
apparently not all of it (e.g. Wielen & Fuchs 1990).